<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b2">I</font> being  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c5">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">I</font>,<font color="Maroon" title="c4">s1</font> holds <br/> <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">I</font>,<font color="Maroon" title="c5">s2</font></span><br/><b>let </b><font color="Maroon" title="c6">a</font> be  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c5">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c4">s1</font> holds <br/> <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c5">s2</font></span><br/><b>let </b><font color="Maroon" title="c7">I</font> be   <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c5">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font> implies  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E41">A1</font></i>: 
 <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c5">s2</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E42">A2</font></i>: 
 <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span><br/>

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">SCMFSA9A:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> ) )</span><br/>

<b>assume </b><a NAME="E3:48"/><i><font color="Green" title="E43">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<a NAME="E4:48"/><i><font color="Green" title="E44">A4</font></i>: 
 <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E2:48"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="scmfsa9a.html#T40" target="_self" title="SCMFSA9A:th.40">Th40</a></i>;<br/>
<a NAME="E5:48"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:48"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="scmfsa6a.html#T38" title="SCMFSA6A:th.38">SCMFSA6A:38</a></i>;<br/>
<a NAME="E6:48"/><b>then </b>
( <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> )
 
<b>by </b><i><a class="txt" href="#E2:48"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
( <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="scmfsa8b.html#T8" title="SCMFSA8B:th.8">SCMFSA8B:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
